∫ calculus

1. **State the problem:** We are asked to find the values of $f(1)$ and the left-hand and right-hand limits of $f(x)$ as $x$ approaches 1, 2, and 3, as well as the two-sided limits

1. **Problem Statement:** Given the piecewise linear function $f(x)$ with specified points and limits, find the values of $f(1)$ and the limits of $f(x)$ as $x$ approaches 1, 2, an

1. **State the problem:** We need to find the integral $$\int \frac{x^2}{3} e^{3x} \, dx$$. 2. **Rewrite the integral:** Factor out the constant \(\frac{1}{3}\):

1. **State the problem:** We need to find the integral $$\int \frac{x^2}{3} e^{3x} \, dx$$. 2. **Rewrite the integral:** Factor out the constant \(\frac{1}{3}\):

1. **State the problem:** Evaluate the double integral $$\int_0^1 \int_y^{\sqrt{y}} (x^2 + y) \, dx \, dy.$$\n\n2. **Understand the integral:** The integral is over $x$ from $y$ to

1. **State the problem:** Evaluate the improper integral $$\int_{-\infty}^{0} \frac{1}{\sqrt[4]{4-x}} \, dx$$ and determine if it converges or diverges. 2. **Rewrite the integral u

1. **State the problem:** Evaluate the integral $$\int_{6}^{9} \frac{\sqrt{x}}{1 + \sqrt[4]{x^3}} \, dx.$$\n\n2. **Rewrite the integral:** Let us express the integrand in terms of

1. The problem asks to determine which option (A, B, or C) gives the best estimate of the area under the curve from $x=0$ to $x=12$ using vertical strips, and then to use that opti

1. **State the problem:** We want to evaluate the double integral

1. **Problem 1:** Evaluate $$\lim_{x \to 0} \frac{x e^x - \ln(1+x)}{x^2}$$ 2. **Recall the expansions:**

1. **Stating the problem:** Evaluate the integral $$\int 8 \cot(x) \cos(x)^2 \ dx$$. 2. **Recall definitions and formulas:**

1. **State the problem:** Solve the differential equation $$\frac{dx}{x^3} = y^4 \, dy$$. 2. **Rewrite the equation:** We want to separate variables to integrate both sides. Rewrit

1. **Problem statement:** Find the area of the region bounded by the curve $y = \cos x$ between $x = 0$ and $x = 2\pi$. 2. **Understanding the problem:** The area bounded by a curv

1. **Problem statement:** We want to find the area of the region bounded by the curve $y = \cos x$ between $x = 0$ and $x = 2\pi$. 2. **Formula used:** The area under a curve $y =

1. **Problem Statement:** Find the derivative $\frac{dy}{dx}$ if $y = e^{\tan^{-1} x}$. 2. **Formula and Rules:** To find $\frac{dy}{dx}$ when $y$ is a function of $x$, use the cha

1. **State the problem:** We want to approximate the area of the region $R$ bounded by the curve $y=\log_{10}x$, the lines $x=2$ and $x=14$, and the $x$-axis using the trapezium ru

1. **State the problem:** Estimate the integral $$\int_5^7 2^{5 - \sqrt{x}} \, dx$$ using the trapezium rule with the given values of $y$ at $x = 5, 5.5, 6, 6.5, 7$.

1. **Problem Statement:** Given the graph of a function $f(x)$, determine the graph of its derivative $f'(x)$. The derivative graph represents the slope of $f(x)$ at each point. 2.

1. **State the problem:** We need to find the limit $$\lim_{x \to \infty} \frac{2 \ln x}{5 x}$$. 2. **Recall the behavior of functions:** As $$x \to \infty$$, $$\ln x$$ grows witho

1. **State the problem:** We need to find the limit $$\lim_{x \to \infty} \frac{\ln x}{2x}$$. 2. **Recall the behavior of functions:** As $x \to \infty$, $\ln x$ grows without boun

1. Probleem: Leiame piirväärtuse $$\lim_{x \to \frac{\pi}{4}} \frac{6 - 6 \cot x}{x - \frac{\pi}{4}}$$ ja määrame väärtused A ja B, kus \(A\) on tuletise arvutamisel saadud funktsi